

Recall, in General Relativity, a heuristic derivation usually begins
with the Newtonian gravity's Poisson equation
\begin{equation}\label{newtonPoisson}
\nabla^{2}\Phi_{N} = 4\pi G_{N}\rho
\end{equation}
where $\Phi_{N}$ is the Newtonian potential, $G_{N}$ is the
gravitation constant, and $\rho$ is the mass-density. It has a
solution of
\begin{equation}\label{generalSolutionSecondOrder}
\Phi_{N}(r) = \frac{c_{0}}{r}
\end{equation}
where $c_{0}$ is constant.
From here, one typically identifies the right hand side of Eq
(\ref{newtonPoisson}) as the time-time component of stress-energy
tensor, and the left hand side is identified as the time-time
component of the Ricci tensor. This is how most approaches to gravity begin.

In conformal gravity, we begin with a different Poisson
equation. Instead of a second order one, we begin with a fourth order
one~\cite{Mannheim:1994ph}
\begin{equation}\label{fourthOrderPoisson}
\nabla^{4}B(r) = f(r)
\end{equation}
which has the general solution for a spherical source
\begin{equation}\label{generalSolutionFourthOrder}
B(r) = \frac{-r}{2}\int^{R}_{0}dxf(x)x^{2} - \frac{1}{6r}\int^{R}_{0}dxf(x)x^{4}.
\end{equation}
Observe that when $r\ll 1$, Eq (\ref{generalSolutionFourthOrder}) has
$c_{1}/r$ be the dominant term and $c_{2}r\to 0$. Thus for small $r$,
we can recover the Newtonian Poisson equation (\ref{newtonPoisson}).

At first, this may be startling to see Eq
(\ref{generalSolutionFourthOrder}) as being proposed for the
gravitational potential. It is counter-intuitive to propose adding an
$\mathcal{O}(r)$ term, we don't observe it at ``small'' scales
(dropping an apples behaves as being in a $\mathcal{O}(1/r)$
potential!). However, at such scales, the potential for
a fourth order Poisson equation behaves as the potential for a second
order one. Additionally, there is observational problems with gravity
that departs from a second order Poisson equation at \emph{large}
distances. So the fourth order approach modifies only what is expected
at \emph{large} distances, and agrees with what we expect at
\emph{small} distances.

Here, we must intervene and confess that there is a terribly strong
no-go theorem: Ostrogradski's theorem (for a beautiful introduction,
see section 2 of Woodard~\cite{Woodard:2006nt}). With
Lagrangians involving terms of second order (or higher) time
derivatives of the position term is unstable. Woodard notes that in
Lagrangians of the form $L(q_{i},\dot{q}_{j},\ddot{q}_{k})$ ``there is
not even any barrier to decay''. Adding insult to injury, the
situation does not improve if we add in higher order derivatives!
